Iteration at the Boundary of Newton Maps

TL;DR

This paper investigates the boundary behavior of holomorphic families of Newton maps of degree at least 3, analyzing their measure limits and rescaling limits through Berkovich dynamics.

Contribution

It provides new estimates for the weak limits of maximal measures and a comprehensive description of rescaling limits for these families.

Findings

Estimate of weak limits of maximal measures

Complete description of rescaling limits

Application of Berkovich dynamics to Newton maps

Abstract

Let $\{N_t\}$ be a holomorphic family of degree $d\ge 3$ Newton maps. By studying the related Berkovich dynamics, we obtain an estimate of the weak limit of the maximal measures of $N_t$. Moreover, we give a complete description of the rescaling limits for $\{N_t\}$.

Figures

Equations

$$P(z)=\prod _{i=1}^d(z-r_i),$$

$$N_P(z)=z-\frac{P(z)}{P^{\prime }(z)}.$$

$$f([X:Y])=[F_a([X:Y]):F_b([X:Y])],$$

$$f([X:Y])=[F_a([X:Y]):F_b([X:Y])]=H_f(X,Y)[f_a([X:Y]):f_b([X:Y])],$$

$$\mu (\{c\})\ge \frac{d_c(f)}{d+d_c(f)},$$

$$\mu (\{\infty \})\ge \frac{1}{2}.$$

$$\mu (\{\infty \})\ge \frac{d}{2d-1}.$$

$${\Psi }_n:{\mathrm{Rat}}_d\to {\mathrm{Rat}}_{d^n},$$

$${\Psi }_n:{\mathbb{P}}^{2d+1}\text{dasharrow}{\mathbb{P}}^{2d^n+1}.$$

$$I(d)=\{f=H_f\hat{f}\in {\mathbb{P}}^{2d+1}:\hat{f}\equiv h\in {\mathbb{P}}^1,H_f(h)=0\}.$$

$$f^n={\Psi }_n(f)=\left(\prod _{k=0}^{n-1}(H_f\circ {\hat{f}}^k{)}^{d^{n-k-1}}\right){\hat{f}}^n.$$

$${\mathcal{C}}_{d,e}:{\mathbb{P}}^{2d+1}\times {\mathbb{P}}^{2e+1}\text{dasharrow}{\mathbb{P}}^{2de+1},$$

$$I(d,e)=\{(f,g)=(H_f\hat{f},H_g\hat{g}):\hat{g}\equiv c\in {\mathbb{P}}^1,H_f(c)=0\}.$$

$${\mathcal{C}}_{d,e}(f,g)=(H_g{)}^d{H}_f(P_g,Q_g)\hat{f}\circ \hat{g},$$

$$f(z)=\frac{z(z+{\mu }_1)}{{\mu }_{2}z+1}$$

$$f([X:Y])=[X^2+{\mu }_{1}XY:{\mu }_{2}XY+Y^2].$$

$$f_t=[1:{\mu }_1(t):0:0:{\mu }_2(t):1]\in {\mathbb{P}}^5$$

$$g([X:Y])=H_g(X,Y)\hat{g}([X:Y])=\{\begin{array}{ll}XY[1:0],& \text{if}{\mu }_1(t)\to \infty ,\\ XY[0:1],& \text{if}{\mu }_1(t)\to 0,\\ (X+cY)[X:Y],& \text{if}{\mu }_1(t)\to c\in \mathbb{C}\setminus \{0\},\end{array}$$

$$f_t([X:Y])\stackrel{\bullet }{\to }\{\begin{array}{ll}[1:0]\text{on}{\mathbb{P}}^1\setminus \{[1:0],[0:1]\}& \text{if}{\mu }_1(t)\to \infty ,\\ [0:1]\text{on}{\mathbb{P}}^1\setminus \{[1:0],[0:1]\}& \text{if}{\mu }_1(t)\to 0,\\ [X:Y]\text{on}{\mathbb{P}}^1\setminus \{[-c:1]\}& \text{if}{\mu }_1(t)\to c\in \mathbb{C}\setminus \{0\}.\end{array}$$

$$P([X:Y])=P_r([X:Y])=[F_a(X,Y):F_b(X,Y)]=\left[\prod _{i=1}^d(X-r_{i}Y):Y^d\right].$$

$$P^{\prime }([X:Y])=P_r^{\prime }([X:Y])=[G_a(X,Y):G_b(X,Y)]=\left[\sum _{j=1}^d\prod _{i\ne j}(X-r_{i}Y):Y^{d-1}\right].$$

$$N_r([X:Y])=N_P([X:Y]):=[X{G}_a(X,Y)-F_a(X,Y):Y{G}_a(X,Y)]\in {\mathbb{P}}^{2d+1}.$$

$$N_r(z)=N_P(z)=z-\frac{P(z)}{P^{\prime }(z)},$$

$$N([X:Y])=Y^m{N}_{\tilde{P}}([X:Y])\in {\mathbb{P}}^{2d+1}.$$

$$N_r([X:Y])=H_{N_r}(X,Y){\hat{N}}_r([X:Y])=XY[2X^2-XY:3XY-2Y^2].$$

$$\mathrm{Fix}(N_r)=\mathrm{Zero}(P_r)\cup \{\infty \}=\{r_1,\cdots ,r_d,\infty \}.$$

$$N_r^{\prime }(z)=\frac{P_r(z)P_r^{\prime \prime }(z)}{(P_r^{\prime }(z){)}^2}.$$

$$\mathrm{Fix}({\hat{N}}_r)=\mathrm{Zero}(P_r)\cup \{\infty \}=\{r_1,\cdots ,r_d,\infty \}.$$

$${\mathrm{NM}}_d\subset {\mathrm{Rat}}_d\subset {\mathbb{P}}^{2d+1}.$$

$$P_r(z)=\prod _{i=1}^d(z-r_i)=\sum _{k=0}^d(-1{)}^k{\sigma }_k{z}^{d-k}$$

Full text

1. Introduction

Let $P$ be a degree $d\geq 2$ monic polynomial with distinct roots. It can be written as

[TABLE]

where $r_{1},\dots,r_{d}$ are distinct complex numbers. The Newton map of $P$ is defined by

[TABLE]

We also write $N_{\{r_{1},\cdots,r_{d}\}}$ to indicate the roots of $P$.

As an algorithm for finding the roots of a polynomial, it is also known as Newton’s method, which is a generally convergent for quadratic polynomials. Cayley [8] noticed the difficulties for Newton’s method to find roots of higher degree polynomials. However, there is a finite universal set such that for any root of any suitably normalized polynomial of degree $d$, there is an initial point in the universal set whose orbit converges to this root under iterations of the corresponding Newton’s method [21]. We refer [19, 30, 31, 43, 47] for more details.

As a dynamical system, Newton map exhibits nice properties. The Julia sets of Newton maps are always connected [44]. Moreover, the cubic Newton maps have locally connected Julia sets except in some very specific case [39]. For the combinatorial properties of Newton maps, every postcritically finite cubic Newton map can be constructed by mating two cubic polynomials [48]. For more dynamical properties of Newton maps, we refer [28, 27, 32]. We also refer [49] for an overview of Newton maps. And there also contains detailed background of known results for Newton maps in [40].

The goal of the present article is to investigate the limiting behaviors of a degenerate holomorphic family of Newton maps. We mainly focus on the weak limits of measures of maximal entropy and the rescaling limits. In the subsequent article [35], we will study the compactifications of the related moduli spaces. Now we set up the statement.

For $d\geq 2$, the space $\mathrm{Rat}_{d}$ of degree $d$ complex rational maps can be naturally identified to a dense open subset of $\mathbb{P}^{2d+1}$. And in the projective coordinates, for $f\in\mathrm{Rat}_{d}$, we can write

[TABLE]

where $F_{a}([X:Y])$ and $F_{b}([X:Y])$ are two relatively prime homogeneous polynomials of degree $d$. Generally, for $f\in\mathbb{P}^{2d+1}$, it determines the coefficients for a pair of homogeneous polynomials of degree $d$. We can write

[TABLE]

where $H_{f}(X,Y)=\mathrm{gcd}(F_{a}([X:Y]),F_{b}([X:Y]))$ and $\hat{f}:=[f_{a}([X:Y]):f_{b}([X:Y])]$ is a rational map of degree at most $d$. The indeterminacy locus $I(d)\subset\mathbb{P}^{2d+1}$ is the collection of $f=H_{f}\hat{f}$, where $\hat{f}$ is a constant in $\mathbb{P}^{1}$ and this constant is a hole of $f$, that is, it is a zero of $H_{f}$.

Following DeMarco [9], we can associate each $f\in\mathbb{P}^{2d+1}$ a probability measure $\mu_{f}$. If $f\in\mathrm{Rat}_{d}$, the measure $\mu_{f}$ is the unique measure such that the measure-theoretic entropy of $f$ attains its maximum. If $f=H_{f}\hat{f}\in\mathbb{P}^{2d+1}\setminus\mathrm{Rat}_{d}$ with $\deg\hat{f}\geq 1$, the measure $\mu_{f}$ is an atomic measure defined on the holes of $f$ and all their preimages under $\hat{f}$. If $f=H_{f}\hat{f}\in\mathbb{P}^{2d+1}\setminus\mathrm{Rat}_{d}$ with $\deg\hat{f}=0$, the measure $\mu_{f}$ is an atomic measure defined on the holes of $f$. Then the map $f\to\mu_{f}$ is continuous if and only if $f\not\in I(d)$ [9, Theorem 0.1].

If we restrict to holomorphic families $\{f_{t}\}$, then function $f_{t}\to\mu_{f_{t}}$ is better behaved. Let $\mathbb{P}^{1}_{\mathrm{Ber}}$ be the Berkovich space over the completion of the field of formal Puiseux series and let $\mathbf{f}:\mathbb{P}^{1}_{\mathrm{Ber}}\to\mathbb{P}^{1}_{\mathrm{Ber}}$ be the associated map for $\{f_{t}\}$. If $f_{t}$ converges to a point in $I(d)$, by regarding the Berkovich dynamical system $(\mathbf{f},\mathbb{P}_{\mathrm{Ber}}^{1})$ as the limit of dynamical systems $(f_{t},\widehat{\mathbb{C}})$ as $t\to 0$, DeMarco and Faber [11, Theorem B] proved the weak limit $\mu$ of measures $\mu_{f_{t}}$ exists and it is a countable sum of atoms. Later they [12] gave a method to compute the measure $\mu$ by introducing and studying what they call an analytically stable pair $(\mathbf{f},\Gamma)$, where $\Gamma\subset\mathbb{P}^{1}_{\mathrm{Ber}}$ is a finite set of type II points. Combining with [9, Theorem 0.1], the measure $\mu$ satisfying

[TABLE]

where $c$ is the constant value of $\hat{f}$ and $d_{c}(f)$ is its depth, that is, $d_{c}(f)$ is the multiplicity of $c$ as a zero of $H_{f}$.

We now apply this to Newton maps. Suppose $\{N_{r(t)}\}$ is a holomorphic family of degree $d\geq 2$ Newton maps with $r(t)=\{r_{1}(t),\cdots,r_{d}(t)\}$. If $N_{r(t)}$ converges to a point $N\in I(d)$, as $t\to 0$, then all $r_{i}(t)$s converge to $z=\infty$, see Lemma 2.15. Thus $z=\infty$ is the unique hole of $N$ and the depth $d_{\infty}(N)=d$. Let $\mu$ be the weak limit of the maximal measures $\mu_{t}$ for $N_{r(t)}$. The inequality (1.1) implies

[TABLE]

But the lower bound in inequality (1.2) is not sharp. Our first result is to give a better lower bound for $\mu(\{\infty\})$.

Theorem 1.1**.**

Let $\{N_{t}\}$ be a holomorphic family of degree $d\geq 2$ Newton maps. Suppose $N_{t}$ converges to a point in $I(d)$. Let $\mu_{t}$ be the maximal measure for $N_{t}$ and let $\mu$ be the weak limit of measures $\mu_{t}$. Then

[TABLE]

In particular, if the Gauss point $\xi_{g}\in\mathbb{P}^{1}_{\mathrm{Ber}}$ is a periodic point for the associated map $\mathbf{N}$ for $\{N_{t}\}$, then $\mu=\delta_{\infty}$.

We point out that in general the lower bound in inequality (1.3) is not sharp either. For example, for quadratic Newton maps, we always have $\mu=\delta_{\infty}$, see Corollary 4.13.

To prove Theorem 1.1, we consider the associated map $\mathbf{N}:\mathbb{P}_{\mathrm{Ber}}^{1}\to\mathbb{P}_{\mathrm{Ber}}^{1}$ for $\{N_{t}\}$. Let $V$ be the set of type II repelling fixed points of $\mathbf{N}$. Consider the convex hull $H_{V}^{\xi_{g}}$ of $V$ and the Gauss point $\xi_{g}$. If the orbit $\mathcal{O}(\xi_{g})$ of $\xi_{g}$ under the map $\mathbf{N}$ does not intersects with $H_{V}^{\xi_{g}}$, we consider the vertices set $\Gamma=\{\xi_{g}\}$ and study the equilibrium $\Gamma$-measures on $\mathbb{P}_{\mathrm{Ber}}^{1}$. Then in this case we can prove $\mu=\delta_{\infty}$. If the orbit $\mathcal{O}(\xi_{g})$ intersects with $H_{V}^{\xi_{g}}$, we construct the vertices sets $\Gamma$ such that the pairs $(\mathbf{N},\Gamma)$ are analytically stable and then apply [12, Theorem C].

Since the indetermiancy locus $I(d)$ is the set of points where the iterate maps $\mathbb{P}^{2d+1}\dasharrow\mathbb{P}^{2d^{n}+1}$ are discontinuous [9, Lemma 2.2], it is interesting to explore the limits of holomorphic families converging to points in $I(d)$ under iterations. More naturally, we work on the moduli space $\mathrm{rat}_{d}:=\mathrm{Rat}_{d}/\mathrm{PGL}_{2}(\mathbb{C})$, modulo the action by conjugation of the group of Möbius transformations.

In quadratic case, the existence of such interesting limits was observed by Stimson [46] in his analysis of the asymptotic behavior of some algebraic curves in $\mathrm{rat}_{2}$. The idea of rescaling limits also appeared in Epstein’s work [14] proving some hyperbolic components are precompact in $\mathrm{rat}_{2}$. Kiwi[25] defined rescaling limits explicitly. For a holomorphic family $\{f_{t}\}$, the rescaling limits arise as limits $M_{t}^{-1}\circ f_{t}^{q}\circ M_{t}\to g$ of rescaled iterates where the convergence is locally uniform outside some finite subset of $\mathbb{P}^{1}$. By studying the associated dynamical system $(\mathbf{f},\mathbb{P}^{1}_{\mathrm{Ber}})$ and relating the rescalings to the periodic repelling type II points in $\mathbb{P}^{1}_{\mathrm{Ber}}$, Kiwi proved for any given holomorphic one-parameter family of degree $d\geq 2$ rational maps, there are at most $2d-2$ dynamically independent rescalings such that the corresponding rescaling limits are not postcritically finite [25, Theorem 1]. Later, applying the trees of spheres, which is based on the Deligne-Mumford compactifications of the moduli spaces of stable curves, and considering the dynamical systems between trees of spheres, Arfeux proved the same results [3, Theorem 1] about the interesting rescaling limits. Moreover, the existence of a periodic sphere for a dynamical cover between trees of spheres corresponds to the existence of a rescaling limit [3, Theorem 2, Theorem 3]. Recently, Arfeux and Cui [2] announced for any arbitrary large integer $n$, there is a holomorphic family in $\mathrm{Rat}_{d}$ for $d\geq 3$ such that it has $n$ dynamically independent rescalings leading to postcritically finite and nonmonomial rescaling limits.

For a holomorphic family of degree $d\geq 3$ Newton maps, we give a complete description for its rescaling limits.

Theorem 1.2**.**

Let $\{N_{t}\}$ be a holomorphic family of degree $d\geq 3$ Newton maps. Then

1. (1)

up to equivalence, $\{N_{t}\}$ has at most $d-1$ rescalings of period $1$. Moreover, each such rescaling leads to a rescaling limit that is conjugate to a degenerated Newton map. 2. (2)

$\{N_{t}\}$* has at most $d-3$ dynamically independent rescalings of periods at least $2$. Moreover, the corresponding rescaling limit for each such rescaling is conjugate to a polynomial of degree at most $2^{d-3}$.*

Let $\mathbf{N}:\mathbb{P}^{1}_{\mathrm{Ber}}\to\mathbb{P}^{1}_{\mathrm{Ber}}$ be the associated map for $\{N_{t}\}$. The period $1$ rescalings for $\{N_{t}\}$ correspond to the type II repelling fixed points of $\mathbf{N}$. To prove part $(1)$, we show the type II repelling fixed points are the points that are the branch points in the convex hull whose endpoints are the type I fixed points of $\mathbf{N}$. To prove $(2)$, we first consider the ramification locus $\mathcal{R}_{\mathbf{N}}$ and give sufficient and necessary conditions for the existence of generalized rescalings of period at least $2$. Then we count the available critical points of $\mathbf{N}$. It will give us the upper bound of the rescalings of higher periods. The main ingredient we use to show the corresponding rescaling limits are conjugate to polynomials is Theorem 1.1. If there exists a rescaling $\{M_{t}\}$ of period $q\geq 2$, then up to dynamical dependence, we can assume $M_{t}$ is affine. By the continuity of the composition maps, $M_{t}^{-1}\circ N_{t}\circ M_{t}$ converges to a point in $I(d)$. We consider the associated map $\widetilde{\mathbf{N}}=\mathbf{M}^{-1}\circ\mathbf{N}\circ\mathbf{M}$ of the holomorphic family $\{M_{t}^{-1}\circ N_{t}\circ M_{t}\}$. Then the Gauss point $\xi_{g}$ is a periodic point of $\widetilde{\mathbf{N}}$. Thus, by Theorem 1.1, the weak limit of the maximal measures of $M_{t}^{-1}\circ N_{t}\circ M_{t}$ is the measure $\delta_{\infty}$. By the invariance of the maximal measures and the continuity of the map $f\to\mu_{f}$ if $f\not\in I(d)$, we know the measure associated to the subalgebraic limit for $M_{t}^{-1}\circ N^{q}_{t}\circ M_{t}$ is $\delta_{\infty}$. Hence, we can show the rescaling limit is a polynomial. We mention here that $(2)$ can also be obtained from the Berkovich dynamics of the map $\mathbf{N}$, see Remark 5.26. Moreover, the subalgrbraic limit of $M_{t}^{-1}\circ N^{q}_{t}\circ M_{t}$ has a unique hole at $\infty$. Therefore, the upper bound of the degree of the rescaling limits implies this subalgrbraic limit is not semistable, see Proposition 5.30.

Outline

In section 2, we recall the relevant backgrounds of Newton maps and prove there is a unique indeterminacy point in boundary of the space of Newton maps, see Proposition 2.18. Section 3 is devoted to describe the basic Berkovich dynamics of Newton maps. It also contains a backgrounds of Berkovich spaces and related dynamics of rational maps. The goal of section 4 is to study the degenerate holomorphic families of Newton maps in measure-theoretic view. We first state DeMarco and Faber’s results about limiting measures, and then prove Theorem 1.1. Finally we prove Theorem 1.2 and related results in section 5. Kiwi’s results are included in this section.

2. Preliminaries

2.1. Iterate Maps on $\mathrm{Rat}_{d}$

For $d\geq 1$, consider the map

[TABLE]

sending $f$ to its $n$-th iterate $f^{n}$. Then $\Psi_{n}$ is proper for all $n\geq 2$ and all $d\geq 2$ [9, Corollary 0.3]. The iterate map $\Psi_{n}$ extends to a rational map

[TABLE]

Define $I(d)\subset\mathbb{P}^{2d+1}$ is the collection of $f=H_{f}\hat{f}$ with $\deg\hat{f}=0$ and the constant value of $\hat{f}$ is a hole of $f$, that is

[TABLE]

The following result, due to DeMarco, claims the indeterminacy loci of the maps $\Psi_{n}$ are $I(d)$ and gives the formula for iteration.

Lemma 2.1**.**

[9, Lemma 2.2]** For $d\geq 1$ and $n\geq 2$, the indeterminacy locus $I(\Psi_{n})$ of the iterate map $\Psi_{n}:\mathbb{P}^{2d+1}\dasharrow\mathbb{P}^{2d^{n}+1}$ is $I(d)$. Moreover, for $f=H_{f}\hat{f}\in\mathbb{P}^{2d+1}\setminus I(d)$,

[TABLE]

More general, for the composition maps, we have

Lemma 2.2**.**

[10, Lemma 2.6]** The composition map

[TABLE]

which sends a pair $(f,g)$ to the composition $f\circ g$ is continuous away from

[TABLE]

Moreover, for $(f,h)=(H_{f}\hat{f},H_{g}\hat{g})\in\mathbb{P}^{2d+1}\times\mathbb{P}^{2e+1}\setminus I(d,e)$,

[TABLE]

where $\hat{g}=[P_{g}:Q_{g}]$.

2.2. Convergence

In this subsection, we consider various notations of convergence of maps in the space of (degenerate) rational maps. Mainly, we define the subalgebraic convergence and show the relations between subalgebraic convergence and locally uniform convergence on the complement of a finite set in $\mathbb{P}^{1}$.

First recall that for $t\in\mathbb{D}$, a collection $\{f_{t}\}\subset\mathbb{P}^{2d+1}$ is a 1-dimensional holomorphic family of degree $d\geq 1$ rational maps if the map $\mathbb{D}\to\mathbb{P}^{2d+1}$, sending $t$ to $f_{t}$, is a holomorphic map such that $f_{t}\in\mathrm{Rat}_{d}$ if $t\not=0$. We say $\{f_{t}\}$ is degenerate if $f_{0}\not\in\mathrm{Rat}_{d}$. A holomorphic family of degree $1$ rational maps is called a moving frame.

Definition 2.3**.**

For a holomorphic family $\{f_{t}\}\subset\mathbb{P}^{2d+1}$, we say $f_{t}$ converges to $f=H_{f}\hat{f}\in\mathbb{P}^{2d+1}$ subalgebraically if $f_{t}$ converges to $f$ in $\mathbb{P}^{2d+1}$ as $t\to 0$. If, in addition, $\deg\hat{f}=d$, we say $f_{t}$ converges to $f\in\mathbb{P}^{2d+1}$ algebraically.

Let’s use the notations $f_{t}\xrightarrow{sa}f$ and $f_{t}\xrightarrow{a}f$ for subalgebraic convergence and algebraic convergence, respectively. Moreover, for a finite subset $S\subset\mathbb{P}^{1}$, denote $f_{t}\xrightarrow{\bullet}\hat{f}$ on $\mathbb{P}^{1}\setminus S$ if $f_{t}$ converges to $f$ locally uniformly on $\mathbb{P}^{1}\setminus S$, and denote $f_{t}\rightrightarrows\hat{f}$ if $f_{t}$ converges to $f$ uniformly on $\mathbb{P}^{1}$.

Lemma 2.4**.**

[25, Lemma 3.2 ]** For a holomorphic $\{f_{t}\}\subset\mathrm{Rat}_{d}$, suppose that $f_{t}\xrightarrow{sa}f=H_{f}\hat{f}$. Then $f_{t}\xrightarrow{\bullet}\hat{f}$ on $\mathbb{P}^{1}\setminus\{h:H_{f}(h)=0\}$.

Conversely, we have

Lemma 2.5**.**

Let $\{f_{t}\}\subset\mathrm{Rat}_{d}$ be a holomorphic family. Suppose that $f_{t}\xrightarrow{\bullet}\hat{f}\in\mathrm{Rat}_{e}$ on $\mathbb{P}^{1}\setminus S$, where $S\subset\mathbb{P}^{1}$ is finite. Then there exists a degree $d-e$ homogeneous polynomial $H$ such that $f_{t}\xrightarrow{sa}H\hat{f}$ with $\{h:H(h)=0\}\subset S$.

Proof.

Suppose that $f_{t}\xrightarrow{sa}g=H_{g}\hat{g}$, as $t\to 0$. Then by Lemma 2.4, we have $f_{t}\xrightarrow{\bullet}\hat{g}$ on $\mathbb{P}^{1}\setminus\{h:H_{g}(h)=0\}$. Thus $\hat{g}=\hat{f}$ on $\mathbb{P}^{1}\setminus(\{h:H_{g}(h)=0\}\cup S)$. So $\hat{g}=\hat{f}$ on $\mathbb{P}^{1}$. We get $\{h:H_{g}(h)=0\}\subset S$. Thus, $g=H_{g}\hat{g}=H_{g}\hat{f}$. ∎

Corollary 2.6**.**

Let $\{f_{t}\}\subset\mathrm{Rat}_{d}$ be a holomorphic family. Then $f_{t}\xrightarrow{a}f$ if and only if $f_{t}\rightrightarrows f$.

We use the following example to illustrate the subalgebraical convergence.

Example 2.7**.**

Quadratic rational maps.

Here we consider the fixed-point normal form for quadratic rational maps [33, Appendix C].

[TABLE]

with $\mu_{1}\mu_{2}\not=1$. Then $z=0$ and $z=\infty$ are fixed points of $f(z)$ with multipliers $\mu_{1}$ and $\mu_{2}$, respectively. And the third fixed point is $z=(1-\mu_{1})/(1-\mu_{2})$ with multiplier $\mu_{3}=(2-\mu_{1}-\mu_{2})/(1-\mu_{1}\mu_{2})$.

In projective coordinates, we can write

[TABLE]

Then $f=[1:\mu_{1}:0:0:\mu_{2}:1]\in\mathbb{P}^{5}$. Consider the holomorphic family $f_{t}$ given by

[TABLE]

with $\mu_{1}(t)\mu_{2}(t)\to 1$. Then $\{f_{t}\}$ is degenerate. Suppose $f_{t}\xrightarrow{sa}g=H_{g}\hat{g}$ as $t\to 0$. We have

[TABLE]

and

[TABLE]

2.3. (Degenerate) Newton Maps

In this subsection, we state some basic background of the (degenerate) Newton maps. For $d\geq 2$, let $P(z)$ be a degree $d$ monic polynomial. Let $r=\{r_{1},\cdots,r_{d}\}$ be the roots of $P(z)$ counted with multiplicity. Then $P(z)$ is uniquely determined by $r$. We may write $P_{r}(z)$ instead of $P(z)$ to emphasis its roots. In projective coordinates, we can write

[TABLE]

Then the derivative $P^{\prime}(z)$ of $P(z)$ can be written as

[TABLE]

Define

[TABLE]

We say $N_{r}$ is a Newton map of degree $d$ if $N_{r}\in\mathrm{Rat}_{d}\subset\mathbb{P}^{2d+1}$. In other word, $N_{r}$ is a Newton map of degree $d$ if and only if $r_{1},\cdots,r_{d}$ are $d$ distinct points in $\mathbb{C}$. In this case, in affine coordinates, we can write $N_{r}:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ by

[TABLE]

and say $N_{r}(z)$ is the Newton map of the polynomial $P_{r}(z)$.

For a point $N\in\mathbb{P}^{2d+1}\setminus\mathrm{Rat}_{d}$, we say $N$ is a degenerate Newton map of degree $d$ if there exists a sequence $\{N_{n}\}$ of Newton maps of degree $d$ such that $N_{n}$ converges to $N$ in $\mathbb{P}^{2d+1}$, as $n\to\infty$. Let $N$ be a degenerate Newton map of degree $d$. Then there exist an integer $d\geq m\geq 0$ and a degree $d-m$ polynomial $\widetilde{P}$ such that

[TABLE]

If $m\not=0$, then $\infty$ is a hole of $N$ with depth $d_{\infty}(N)=m$. All other holes of $N$ are the multiple roots $r_{i}$ of $\widetilde{P}$ and the corresponding depths $d_{r_{i}}(N)$ are $m_{\widetilde{P}}(r_{i})-1$, where $m_{\widetilde{P}}(r_{i})$ is the multiplicity of $r_{i}$ as a zero of $\widetilde{P}$. Note if $m=0$, then $\widetilde{P}$ is a degree $d$ polynomial with multiple roots, and the formula (2.3) coincides to formula (2.1).

In general, let $r$ be a set of $d$ points (not necessary distinct) in $\widehat{\mathbb{C}}$. Suppose there are $m\geq 0$ many $\infty$s in $r$. Let $\widetilde{P}$ be the polynomial of degree $d-m$ whose roots coincide to the points in $r$ that are not $\infty$. Then we define $N_{r}$ to be the (degenerate) Newton map $N_{r}$ by formula (2.3).

Example 2.8**.**

Let $r=\{0,0,1,\infty\}$. Then $N_{r}$ has holes at [math] and $\infty$, and each hole has depth $1$. More precisely,

[TABLE]

Note here we have $\widehat{N}_{r}=\widehat{N}_{\{0,0,1,\infty\}}=\widehat{N}_{\{0,0,1\}}$.

For $f\in\mathrm{Rat}_{d}$, denote by $\mathrm{Fix}(f)$, $\mathrm{Zero}(f)$ and $\mathrm{Crit}(f)$ for the set of fixed points, the set of zeros and the set of critical points of $f$, respectively. Now we state some basic facts about the Newton map $N_{r}(z)$.

Proposition 2.9**.**

For $d\geq 2$, let $r=\{r_{1},\cdots,r_{d}\}$ be a set of $d$ distinct points in $\mathbb{C}$, and let $N_{r}$ be the Newton map of the polynomial $P_{r}$. Then

1. (1)

The set of fixed points of $N_{r}$ is

[TABLE]

All $r_{i}$’s are superattracting fixed points and $z=\infty$ is the unique repelling fixed point. Moreover, the multiplier of $N_{r}$ at $z=\infty$ is $d/(d-1)$. 2. (2)

The derivative of $N_{r}$ is

[TABLE]

In particular, $\mathrm{Crit}(N_{r})=\mathrm{Zero}(P_{r})\cup\mathrm{Zero}(P^{\prime\prime}_{r}).$

Remark 2.10**.**

If $N_{r}$ is a degenerate Newton map, then we still have

[TABLE]

But $r_{i}$ may not be a superattracting fixed point. In fact, it is an attracting fixed point with multiplier $(m_{P_{r}}(r_{i})-1)/d$, where $m_{P_{r}}(r_{i})$ is the multiplicity of $r_{i}$ as a zero of $P_{r}$. Moreover, the depth $d_{r_{i}}(N_{r})=m_{P_{r}}(r_{i})-1$.

For a Newton map $N_{r}$ of degree $d$ with $r=\{r_{1},\cdots,r_{d}\}$, at each $r_{i}$, there is a neighborhood $U_{i}$ such that for each $z\in U_{i}$, $N^{k}_{r}(z)\to r_{i}$ as $k\to\infty$. The B$\ddot{\text{o}}$ttcher’s theorem [34, Theorem 9.1] implies that this convergence is at least quadratic. We say a critical point $c\in\mathrm{Crit}(N_{r})$ is free if $c\in\mathrm{Zero}(P^{\prime\prime}_{r})\setminus\mathrm{Zero}(P_{r})$. If $N_{r}$ has an attracting cycle of period at least $2$, then at least one free critical point is attracted to this cycle.

In the case $d=2$, we can conjugate $N\in\mathrm{NM}_{2}$ by some $M\in\mathrm{PGL}_{2}(\mathbb{C})$ such that the two superattracting fixed points of $M^{-1}\circ N\circ M$ are [math] and $\infty$. This fact is known by Cayley, see [1] for more history. Thus, we have

Proposition 2.11**.**

Any quadratic Newton map $N_{r}(z)$ is conjugate to $z\to z^{2}$.

2.4. The Space $\mathrm{NM}_{d}$

For $d\geq 2$, denote by $\mathrm{NM}_{d}$ the set of degree $d$ Newton maps. Then

[TABLE]

Let $\overline{\mathrm{NM}}_{d}\subset\mathbb{P}^{2d+1}$ be the closure of $\mathrm{NM}_{d}$ in $\mathbb{P}^{2d+1}$.

Let $r=\{r_{1},\cdots,r_{d}\}\subset\mathbb{C}$ be a set of $d$ distinct points. Then

[TABLE]

and

[TABLE]

where the $\sigma_{j}$’s are the elementary symmetric functions of $r_{i}$’s. In projective coordinates, we can express the Newton map $N_{r}$ as following

[TABLE]

Remark 2.12**.**

1. (1)

The formula (2.4) of $N_{r}([X:Y])$ also works for the degenerate Newton map of degree $d$ for which $\infty$ is not a hole. 2. (2)

In general, let $r=\{r_{1},\cdots,r_{d}\}$ be the set of $d$ points (not necessary distinct) in $\widehat{\mathbb{C}}$. Suppose there are $d_{\infty}$ many $r_{i}$s that are $\infty$. Then we can write $N_{r}([X:Y])=Y^{d_{\infty}}N_{\tilde{r}}([X:Y])$, where $\tilde{r}\subset r$ is the subset consisting of $r_{i}\in\mathbb{C}$. The formula 2.4 gives the projective coordinate of $N_{\tilde{r}}([X:Y])$. Hence, we can get the projective coordinates of $N_{r}$.

Since a holomorphic family of rational maps is a family in which the coefficients may be expressed as holomorphic functions of the parameter $t$, the locations of the fixed points are thus algebraic functions of this parameter. It is well-known that algebraic functions of one complex variable t are given by Puiseux series in $t$, see [41]. Thus by a change in the parameterization of the form $t=s^{k}$ we may assume that the fixed points themselves are holomorphic functions of one parameter. In summary, when considering holomorphic families of Newton maps, it suffices for our purposes to consider holomorphic families where the roots $r_{i}(t)$ are holomorphic functions of $t$.

Now we state an easy lemma to show the subalgebraic limit of a holomorphic family $\{N_{r(t)}\}$ of Newton maps.

Lemma 2.13**.**

Let $\{N_{r(t)}(z)\}$ be a holomorphic family of degree $d\geq 2$ Newton maps with $r(t)=\{r_{1}(t),\cdots,r_{d}(t)\}$. Suppose $r_{i}(t)\to r_{i}\in\widehat{\mathbb{C}}$ as $t\to 0$, then in projective coordinates,

[TABLE]

where $r=\{r_{1},\cdots,r_{d}\}$.

We use the following example to illustrate Lemma 2.13. And we draw the corresponding dynamics plane by the software “fractalstream”.

Example 2.14**.**

1. (1)

Let $r(t)=\{0,1,i/t\}$. Consider the Newton map $N_{r(t)}$. As $t\to 0$, we have

[TABLE] 2. (2)

Let $\tilde{r}(t)=\{0,1,ti\}$. Consider the Newton map $N_{\tilde{r}(t)}$. As $t\to 0$, we have

[TABLE]

Recall $I(d)\subset\mathbb{P}^{2d+1}$ is the indeterminacy locus. Then

Lemma 2.15**.**

Let $\{N_{r(t)}\}$ be a holomorphic family of degree $d\geq 2$ Newton maps. Suppose $N_{r(t)}\xrightarrow{sa}N=H_{N}\widehat{N}$ as $t\to 0$. Then $N\in I(d)$ if and only if $r_{i}(t)\to\infty$ for all $i=1,\cdots,d$.

Proof.

If $r_{i}(t)\to\infty$ for $i=1,\cdots,d$, then as $t\to 0$, we have $\sigma_{d}(t)\to\infty$ and for $j=1,\cdots,d-1$,

[TABLE]

Hence, by formula (2.4), as $t\to 0$,

[TABLE]

Thus, $N([X:Y])=[Y^{d}:0]=Y^{d}[1:0]\in I(d)$.

Conversely, let $E=\{i:r_{i}(t)\not\to\infty\}$. By Lemma 2.13, $\#E\leq 1$ since $\deg\widehat{N}=0$. If $\#E=1$, suppose $i\in E$ and $r_{i}(t)\to a\in\mathbb{C}$, as $t\to 0$. Then $f([X:Y])=Y^{d}[a:1]$. Thus, in this case $N\not\in I(d)$. So $\#E=0$, that is, all $r_{i}(t)\to\infty$, as $t\to 0$. ∎

Corollary 2.16**.**

If $N=H_{N}\widehat{N}\in\partial\overline{\mathrm{NM}}_{d}$ with $\deg\widehat{N}=0$, then $\mathrm{Hole}(N)=\{\infty\}$.

Corollary 2.17**.**

Let $\{N_{r(t)}\}$ be a holomorphic family of degree $d\geq 2$ Newton maps with $r(t)=\{r_{1}(t),\cdots,r_{d}(t)\}$. Suppose $r_{i_{0}}(t)\not\to\infty$ for some $1\leq i_{0}\leq d$ as $t\to 0$, and suppose $N_{r(t)}\xrightarrow{sa}N=H_{N}\widehat{N}$. Then for any $q\geq 1$, $N^{q}_{r(t)}\xrightarrow{sa}N^{q}$ and hence $N^{q}_{r(t)}\xrightarrow{\bullet}\widehat{N}^{q}$ outside a finite set.

Proof.

From Lemma 2.15, we have $f\not\in I(d)$. It follows the Lemmas 2.1 and 2.4. ∎

Lemma 2.15 implies immediately the following fundamental result.

Proposition 2.18**.**

For $d\geq 2$,

[TABLE]

3. Berkovich Dynamics

3.1. Berkovich Space and Related Dynamics

In this subsection, we summarize the definitions and main properties of Berkovich spaces and rational maps on Berkovich spaces used in this work. For more details, we refer [4, 6, 5, 22, 37, 38].

Let $\mathbb{C}\{\{t\}\}$ be the field of formal Puiseux series over $\mathbb{C}$. Then $\mathbb{C}\{\{t\}\}$ can be equipped with a non-Archimedean absolute value $|\cdot|_{\mathbb{C}\{\{t\}\}}$ by defining

[TABLE]

where $n_{0}$ is the smallest integer $n\in\mathbb{Z}$ such that $a_{n}\not=0$. For more details of non-Archimedean fields, we refer [13]. Then the field $\mathbb{C}\{\{t\}\}$ is algebraically closed but not complete. Let $\mathbb{L}$ be the completion of the field $\mathbb{C}\{\{t\}\}$, see [23, 24, 25]. It consists of all formal sums of the form $\sum_{n\geq 0}a_{n}t^{q_{n}}$, where $\{q_{n}\}$ is a sequence of rational numbers increasing to $\infty$ and $a_{n}\in\mathbb{C}$. Let $|\cdot|_{\mathbb{L}}$ be the extension of the non-Archimedean absolute value $|\cdot|_{\mathbb{C}\{\{t\}\}}$ so that $|a_{0}t^{q_{0}}+\cdots|_{\mathbb{L}}=e^{-q_{0}}$. To abuse of notations, we may write $|\cdot|$ for $|\cdot|_{\mathbb{L}}$. Then the ring of integers of the field $\mathbb{L}$ is

[TABLE]

and the unique maximal ideal $\mathcal{M}_{\mathbb{L}}$ of $\mathcal{O}_{\mathbb{L}}$ consists of series with zero constant term, i.e.

[TABLE]

The residue field $\mathcal{O}_{\mathbb{L}}/\mathcal{M}_{\mathbb{L}}$ is canonically isomorphic to $\mathbb{C}$.

Let $|\mathbb{L}^{\times}|$ be the value group of $\mathbb{L}$. Given $a\in\mathbb{L}$ and $r>0$, define

[TABLE]

If $r\in|\mathbb{L}^{\times}|$, we say that $D(a,r)$ is an open rational disk in $\mathbb{L}$ and $\overline{D}(a,r)$ is a closed rational disk in $\mathbb{L}$. If $r\not\in|\mathbb{L}^{\times}|$, then $D(a,r)=\overline{D}(a,r)$. And we call it an irrational disk. Let $U(a,r)\subset\mathbb{L}$ be a disk centered at $a\in\mathbb{L}$ with radius $r>0$, that is, $U$ has the form $D(a,r)$ or $\overline{D}(a,r)$. Then if $b\in U(a,r)$, we have $U(a,r)=U(b,r)$. Moreover, the radius $r$ is the same as the diameter of $U(a,r)$, that is $r=\sup\{|z-w|,w\in U(a,r)\}$. Furthermore, if two disks have a nonempty intersection, then one must contain the other. Finally, we should mention here every disk in $\mathbb{L}$ is both open and closed under the topology of $\mathbb{L}$.

The Berkovich affine line $\mathbb{A}_{\mathrm{Ber}}^{1}(\mathbb{L})$ is the set of all multiplicative seminorms on the ring $\mathbb{L}[z]$ of polynomials over $\mathbb{L}$, whose restriction to the field $\mathbb{L}\subset\mathbb{L}[z]$ is equal to the given absolute value $|\cdot|$. To ease notation, we may write $\mathbb{A}_{\mathrm{Ber}}^{1}$ for $\mathbb{A}_{\mathrm{Ber}}^{1}(\mathbb{L})$. For $a\in\mathbb{L}$ and $r\geq 0$, let $\xi_{a,r}$ be the seminorm defined by

[TABLE]

for $f\in\mathbb{L}[z]$. Then there are $4$ types of points in $\mathbb{A}_{\mathrm{Ber}}^{1}$:

1. Type I. $\xi_{a,0}$ for some $a\in\mathbb{L}$.

2. Type II. $\xi_{a,r}$ for some $a\in\mathbb{L}$ and $r\in|\mathbb{L}^{\times}|$.

3. Type III. $\xi_{a,r}$ for some $a\in\mathbb{L}$ and $r\notin|\mathbb{L}^{\times}|$.

4. Type IV. A limit of seminorms $\{\xi_{a_{i},r_{i}}\}_{i\geq 0}$, where the corresponding sequence of closed disks $\{\overline{D}_{a_{i},r_{i}}\}_{i\geq 0}$ satisfies $\overline{D}_{a_{i+1},r_{i+1}}\subset\overline{D}_{a_{i},r_{i}}$ and $\bigcap\limits_{i}\overline{D}_{a_{i},r_{i}}=\emptyset$.

We can identify $\mathbb{L}$ with the type I points in $\mathbb{A}_{\mathrm{Ber}}^{1}$ via $a\to\xi_{a,0}$. The point $\xi_{0,1}\in\mathbb{A}_{\mathrm{Ber}}^{1}$ is called the Gauss point and denoted by $\xi_{g}$. We put the Gelfand topology (weak topology) on $\mathbb{A}_{\mathrm{Ber}}^{1}$, which makes the map $\mathbb{A}_{\mathrm{Ber}}^{1}\to[0,+\infty)$, sending $\xi$ to $|f|_{\xi}$, continuous for each $f\in\mathbb{L}[z]$. Then $\mathbb{A}_{\mathrm{Ber}}^{1}$ is locally compact, Hausdorff and uniquely path-connected.

For $\xi,\xi^{\prime}\in\mathbb{A}^{1}_{\mathrm{Ber}}$, we say $\xi\leq\xi^{\prime}$ if $|f|_{\xi}\leq|f|_{\xi^{\prime}}$ for all polynomials $f\in\mathbb{L}[z]$. For types I, II and III points, $\xi_{a_{1},r_{1}}\leq\xi_{a_{2},r_{2}}$ if and only if $\overline{D}(a_{1},r_{1})\subset\overline{D}(a_{2},r_{2})$. Then $\leq$ is a partial order on $\mathbb{A}^{1}_{\mathrm{Ber}}$. Denote by $\xi\vee\xi^{\prime}$ the least upper bound with respect to the partial order $\leq$. Then $\xi\vee\xi^{\prime}$ always exists and is unique. The small metric on $\mathbb{A}_{\mathrm{Ber}}^{1}$ is defined by

[TABLE]

where $\mathrm{diam}(\xi)$ is the affine diameter of the point $\xi$, that is, if $\{\xi_{a_{i},r_{i}}\}$ is a sequence of seminorms converging to $\xi$, then $\mathrm{diam}(\xi)=\lim\limits_{i\to\infty}r_{i}$. The topology on $\mathbb{A}_{\mathrm{Ber}}^{1}$ induced by the metric $d$ is called the strong topology. It is strictly finer than the weak topology. The Berkovich hyperbolic space is defined by

[TABLE]

Then we define the path distance metric $\rho$ on $\mathbb{H}_{\mathrm{Ber}}^{1}$ by

[TABLE]

The restriction of the strong topology to $\mathbb{H}_{\mathrm{Ber}}^{1}$ coincides with the metric topology induced by $\rho$. The space $\mathbb{H}_{\mathrm{Ber}}^{1}$ is complete for this metric, but not locally compact.

The Berkovich projective line $\mathbb{P}_{\mathrm{Ber}}^{1}$ is obtained by gluing two copies of $\mathbb{A}_{\mathrm{Ber}}^{1}$ along $\mathbb{A}_{\mathrm{Ber}}^{1}\setminus\{0\}$ via the map $\xi\to 1/\xi$. Then we can associate the Gelfand topology and strong topology on $\mathbb{P}_{\mathrm{Ber}}^{1}$. Under the Gelfand topology, The Berkovich projective line $\mathbb{P}_{\mathrm{Ber}}^{1}$ is a compact, Hausdorff, uniquely path-connected topological space and contains the projective line $\mathbb{P}^{1}_{\mathbb{L}}$ as a dense subset.

The space $\mathbb{P}_{\mathrm{Ber}}^{1}$ has tree structure. For a point $\xi\in\mathbb{P}_{\mathrm{Ber}}^{1}$, we can define an equivalence relation on $\mathbb{P}_{\mathrm{Ber}}^{1}\setminus\{\xi\}$, that is, $\xi^{\prime}$ is equivalent to $\xi^{\prime\prime}$ if $\xi^{\prime}$ and $\xi^{\prime\prime}$ are in the same connected component of $\mathbb{P}_{\mathrm{Ber}}^{1}\setminus\{\xi\}$. Such an equivalence class $\vec{v}$ is called a direction at $\xi$. We say that the set $T_{\xi}\mathbb{P}_{\mathrm{Ber}}^{1}$ formed by all directions at $\xi$ is the tangent space at $\xi$. For $\vec{v}\in T_{\xi}\mathbb{P}_{\mathrm{Ber}}^{1}$, denote by $\mathbf{B}_{\xi}(\vec{v})^{-}$ the component of $\mathbb{P}_{\mathrm{Ber}}^{1}\setminus\{\xi\}$ corresponding to the direction $\vec{v}$. If $\xi\in\mathbb{P}_{\mathrm{Ber}}^{1}$ is a type I or IV point, $T_{\xi}\mathbb{P}_{\mathrm{Ber}}^{1}$ consists of a single direction. If $\xi\in\mathbb{P}_{\mathrm{Ber}}^{1}$ is a type III point, $T_{\xi}\mathbb{P}_{\mathrm{Ber}}^{1}$ consists of two directions. If $\xi\in\mathbb{P}_{\mathrm{Ber}}^{1}$ is a type II point, the directions in $T_{\xi}\mathbb{P}_{\mathrm{Ber}}^{1}$ are in one-to-one correspondence with the elements in $\mathbb{P}^{1}$. Since the Gauss point $\xi_{g}$ is a type II point, we can identify $T_{\xi_{g}}\mathbb{P}_{\mathrm{Ber}}^{1}$ to $\mathbb{P}^{1}$ by the correspondence $T_{\xi_{g}}\mathbb{P}_{\mathrm{Ber}}^{1}\to\mathbb{P}^{1}$ sending $\vec{v}_{x}$ to $x$, where $\vec{v}_{x}$ is the direction at $\xi_{g}$ such that $\mathbf{B}_{\xi_{g}}(\vec{v}_{x})^{-}$ contains all the type I points whose images are $x$ under the canonical reduction map $\mathbb{P}^{1}_{\mathbb{L}}\to\mathbb{P}^{1}$.

The spherical metric on the projective $\mathbb{P}^{1}_{\mathbb{L}}$ is defined as follows: for points $z=[x:y]$ and $w=[u:v]$ in $\mathbb{P}^{1}_{\mathbb{L}}$,

[TABLE]

Equivalently,

[TABLE]

Recall a degree $d\geq 1$ rational map $\phi:\mathbb{P}^{1}_{\mathbb{L}}\to\mathbb{P}^{1}_{\mathbb{L}}$ is represented by a pair $\phi_{a},\phi_{b}\in\mathbb{L}[X,Y]$ of degree $d$ homogeneous polynomials with no common factors, that is, $\phi([X:Y])=[\phi_{a}(X,Y):\phi_{b}(X,Y)]$ for all $[X:Y]\in\mathbb{P}^{1}_{\mathbb{L}}$. Equivalently, the map $\phi$ can be considered as the quotient of two relatively prime polynomials, of which the greatest degree is $d$. Then $\phi$ induces a map from $\mathbb{P}_{\mathrm{Ber}}^{1}$ to itself. We use the same notation $\phi$ for the induced map. For the Gauss point $\xi_{g}\in\mathbb{P}_{\mathrm{Ber}}^{1}$, the corresponding closed disk $\overline{D}(0,1)\subset\mathbb{L}$ is a disjoint union of open disks $D(a,1)$ for $a\in\mathbb{C}$. Note for all but finitely many such $a$, we have $\phi(D(a,1))$ is an open disk $D(\phi(a),r)$ for some $r\in|\mathbb{L}^{\times}|$. Then $\phi(\xi_{g})$ is the point $\xi_{\phi(a),r}\in\mathbb{P}_{\mathrm{Ber}}^{1}$ for any such choice of $a$. For an arbitrary type II point $\xi_{b,s}$, pick $M\in\mathrm{PGL}_{2}(\mathbb{L})$ such that $M(\xi_{g})=\xi_{b,r}$. Then apply the previous discussion to $\phi\circ M$ and $\phi(\xi_{b,s})=\phi\circ M(\xi_{g})$. Since type II points are dense in $\mathbb{P}_{\mathrm{Ber}}^{1}$, we can get the image $\phi(\xi)$ for any $\xi\in\mathbb{P}_{\mathrm{Ber}}^{1}$.

For a holomorphic family

[TABLE]

of degree $d$ rational maps, let $\mathbf{a_{d}},\cdots,\mathbf{a_{0}},\mathbf{b_{d}},\cdots,\mathbf{b_{0}}$ be the power series expressions of the coefficients $a_{d}(t),\cdots,a_{0}(t),b_{d}(t),\cdots,b_{0}(t)$, respectively. Then the degree $d$ rational map $\mathbf{f}:\mathbb{P}^{1}_{\mathbb{L}}\to\mathbb{P}^{1}_{\mathbb{L}}$ given by

[TABLE]

is called, following Kiwi [25], the rational map associated to $\{f_{t}\}$. Hence $\{f_{t}\}$ induces a map $\mathbf{f}:\mathbb{P}_{\mathrm{Ber}}^{1}\to\mathbb{P}_{\mathrm{Ber}}^{1}$.

Let $P(z)\in\mathcal{O}_{\mathbb{L}}[z]$ be a polynomial and let $\mathrm{Red}(P)$ be the image of $P(z)$ in $\mathcal{O}_{\mathbb{L}}[z]/\mathcal{M}_{\mathbb{L}}[z]$. We say $\mathrm{Red}(P)$ is the reduction of $P$. For a rational map $\phi=P/Q\in\mathbb{L}(z)$, where $P,Q\in\mathbb{L}[z]$ with no common zeros. We can normalize $\mathbf{\phi}$ such that $P,Q\in\mathcal{O}_{\mathbb{L}}(z)$ and the maximal absolute value of coefficients of $P$ and $Q$ is $1$. For a normalized rational map $\phi=P/Q\in\mathbb{L}(z)$, the reduction $\mathrm{Red}(\phi)$ is defined by

[TABLE]

In fact, if $\phi$ is induced from a holomorphic family, the reduction $\mathrm{Red}(\phi)$ coincides with the limit of $\phi(z)$ as $t\to 0$. For convenience, for an arbitrary rational map $\phi\in\mathbb{L}(z)$, we write $\mathrm{Red}(\phi)$ for the reduction of the normalization of $\phi$.

Lemma 3.1**.**

Let $\mathbf{\phi}(z),\mathbf{\psi}(z)\in\mathbb{L}(z)$ be two rational maps. Then

1. (1)

$\mathrm{Red}(\phi\cdot\psi)=\mathrm{Red}(\phi)\cdot\mathrm{Red}(\psi)$, 2. (2)

$\mathrm{Red}(\phi+\psi)=\mathrm{Red}(\phi)+\mathrm{Red}(\psi)$, 3. (3)

If $\deg\mathrm{Red}(\psi)\geq 1$, then $\mathrm{Red}(\phi\circ\psi)=\mathrm{Red}(\phi)\circ\mathrm{Red}(\psi)$.

The following result, originally proved by Rivera-Letelier, gives a characteristic of rational maps with nonconstant reductions.

Proposition 3.2**.**

[4, Corollary 9.27]** Let $\mathbf{\phi}(z)\in\mathbb{L}(z)$ be a normalized rational map. Then $\deg\mathrm{Red}(\phi)\geq 1$ if and only if $\phi(\xi_{g})=\xi_{g}$.

At each point $\xi\in\mathbb{P}^{1}_{\mathrm{Ber}}$, denote by $\deg_{\xi}\phi$ the local degree of $\phi$ at $\xi$. Then

Lemma 3.3**.**

[4, Section 9.1]** Let $\mathbf{\phi}(z)\in\mathbb{L}(z)$ be a nonconstant rational map. Let $\xi\in\mathbb{P}^{1}_{\mathrm{Ber}}$ and $\vec{v}\in T_{\xi}\mathbb{P}^{1}_{\mathrm{Ber}}$. Then there is a positive integer $m$ and a point $\xi^{\prime}\in\mathbf{B}_{\xi}(\vec{v})^{-}$ such that

1. (1)

$\deg_{\xi^{\prime\prime}}\phi=m$* for all $\xi^{\prime\prime}\in(\xi,\xi^{\prime})$, and* 2. (2)

$\rho(\phi(\xi),\phi(\xi^{\prime\prime}))=m\rho(\xi,\xi^{\prime\prime})$* for all $\xi^{\prime\prime}\in(\xi,\xi^{\prime})$.*

The integer $m$ is called the directional multiplicity and denoted by $m_{\phi}(\xi,\vec{v})$.

Let $\xi\in\mathbb{P}^{1}_{\mathrm{Ber}}$. For any $\vec{v}\in T_{\xi}\mathbb{P}_{\mathrm{Ber}}^{1}$, there is a unique $\vec{w}\in T_{\phi(\xi)}\mathbb{P}_{\mathrm{Ber}}^{1}$ such that for any $\xi^{\prime}$ sufficiently near $\xi$, $\phi(\xi^{\prime})\in\mathbf{B}_{\phi(\xi)}(\vec{w})^{-}$. Thus the rational map $\phi$ induces a map

[TABLE]

sending the direction $\vec{v}$ to the corresponding direction $\vec{w}$.

The following lemma gives the relations between local degrees and directional multiplicities.

Lemma 3.4**.**

[4, Theorem 9.22]** Let $\phi\in\mathbb{L}(z)$ be a nonconstant rational map. Let $\xi\in\mathbb{P}^{1}_{\mathrm{Ber}}$. Then

1. (1)

for each tangent $\vec{w}\in T_{\phi(\xi)}\mathbb{P}^{1}_{\mathrm{Ber}}$, we have

[TABLE] 2. (2)

the induced map $\phi_{\ast}:T_{\xi}\mathbb{P}_{\mathrm{Ber}}^{1}\to T_{\phi(\xi)}\mathbb{P}_{\mathrm{Ber}}^{1}$ is surjective. If $\xi$ is of type I, III, or IV, then $\deg_{\xi}\phi=m_{\phi}(\xi,\vec{v})$ for each $\vec{v}\in T_{\xi}\mathbb{P}_{\mathrm{Ber}}^{1}$.

The following result allows us to compute the local degree of $\phi$ at a type II point.

Proposition 3.5**.**

[36, Proposition 3.3]** Let $\phi\in\mathbb{L}(z)$ be a nonconstant rational function, and let $\xi\in\mathbb{P}^{1}_{\mathrm{Ber}}$ be a type II point. Set $\xi^{\prime}=\phi(\xi)$ and choose $M_{1},M_{2}\in\mathrm{PGL}_{2}(\mathbb{L})$ such that $M_{1}(\xi_{g})=\xi$ and $M_{2}(\xi_{g})=\xi^{\prime}$. Let $\psi=M_{2}^{-1}\circ\phi\circ M_{1}$. Then $\deg\mathrm{Red}(\psi)\geq 1$ and

[TABLE]

For each $a\in\mathbb{P}^{1}$, if $\vec{v}_{a}\in T_{\xi}\mathbb{P}^{1}_{\mathrm{Ber}}$ is the associated tangent direction under the bijection between $T_{\xi}\mathbb{P}^{1}_{\mathrm{Ber}}$ and $\mathbb{P}^{1}$ afforded by $(M_{1}^{-1})_{\ast}$, we have

[TABLE]

Now let’s state a fundamental fact in Berkovich dynamics.

Proposition 3.6**.**

[36, Lemma 2.1]** Let $\phi:\mathbb{P}_{\mathrm{Ber}}^{1}\to\mathbb{P}_{\mathrm{Ber}}^{1}$ be a rational map of degree at least $1$. For $\xi\in\mathbb{P}_{\mathrm{Ber}}^{1}$ and $\vec{v}\in T_{\xi}\mathbb{P}_{\mathrm{Ber}}^{1}$, the image $\phi(\mathbf{B}_{\xi}(\vec{v})^{-})$ always contains $\mathbf{B}_{\phi(\xi)}(\phi_{\ast}(\vec{v}))^{-}$, and either $\phi(\mathbf{B}_{\xi}(\vec{v})^{-})=\mathbf{B}_{\phi(\xi)}(\phi_{\ast}(\vec{v}))^{-}$ or $\phi(\mathbf{B}_{\xi}(\vec{v})^{-})=\mathbb{P}_{\mathrm{Ber}}^{1}$. Moreover

1. (1)

if $\phi(\mathbf{B}_{\xi}(\vec{v})^{-})=\mathbf{B}_{\phi(\xi)}(\phi_{\ast}(\vec{v}))^{-}$, then each $\xi^{\prime}\in\mathbf{B}_{\phi(\xi)}(\phi_{\ast}(\vec{v}))^{-}$ has $m_{\phi}(\xi,\vec{v})$ preimages in $\mathbf{B}_{\xi}(\vec{v})^{-}$, counting multiplicities; 2. (2)

if $\phi(\mathbf{B}_{\xi}(\vec{v})^{-})=\mathbb{P}_{\mathrm{Ber}}^{1}$, there is an integer $s_{\phi}(\xi,\vec{v})>0$ such that each $\xi^{\prime}\in\mathbf{B}_{\phi(\xi)}(\phi_{\ast}(\vec{v}))^{-}$ has $m_{\phi}(\xi,\vec{v})+s_{\phi}(\xi,\vec{v})$ preimages in $\mathbf{B}_{\xi}(\vec{v})^{-}$ and each $\xi^{\prime\prime}\in\mathbb{P}_{\mathrm{Ber}}^{1}\setminus\mathbf{B}_{\phi(\xi)}(\phi_{\ast}(\vec{v}))^{-}$ has $s$ preimages in $\mathbf{B}_{\xi}(\vec{v})^{-}$, counting multiplicities.

Following Faber [15], the integer $s_{\phi}(\xi,\vec{v})$ is called the surplus multiplicity.

The following lemma gives us a criterion for which Berkovich disk $\mathbf{B}_{\xi}(\vec{v})^{-}$ is mapped to another Berkovich disk.

Lemma 3.7**.**

[4, Theorem 9.42]** Let $\phi\in\mathbb{L}(z)$ be a nonconstant rational map. Then $\phi(\mathbf{B}_{\xi}(\vec{v})^{-})$ is a Berkovich disk if and only if for each $c\in\mathbf{B}_{\xi}(\vec{v})^{-}$, the local degree of $\phi$ is nonincreasing on the directed segment $[\xi,c]$.

3.2. Berkovich Dynamics of Newton maps

Fix $d\geq 3$. Let

[TABLE]

be a set of $d$ distinct points. Set $P_{r}(z)=\prod_{i=1}^{d}(z-r_{i})\in\mathbb{L}[z]$. Same as complex case, the Newton map $N_{r}:\mathbb{P}^{1}_{\mathbb{L}}\to\mathbb{P}^{1}_{\mathbb{L}}$ of the polynomial $P_{r}$ is defined by

[TABLE]

The map $N_{r}$ extends to a map $N_{r}:\mathbb{P}^{1}_{\mathrm{Ber}}\to\mathbb{P}^{1}_{\mathrm{Ber}}$. To ease notation, we write $N$ for $N_{r}$. In this subsection, we state some fundamental properties of $N$ in the Berkovich space $\mathbb{P}^{1}_{\mathrm{Ber}}$. These properties shed light on the case when we consider holomorphic families of Newton maps. Indeed, if $\{N_{r(t)}\}$ is a holomorphic family of degree $d$ Newton maps with $r(t)=\{r_{1}(t),\cdots,r_{d}(t)\}$, by regarding each $r_{i}(t)$ as an element in $\mathbb{L}$, the associated map for $\{N_{r(t)}\}$ is a Newton map in $\mathbb{L}(z)$.

Since $\mathbb{L}$ is a complete and algebraically closed field of characteristic zero, the map $N$ has $2d-2$ critical points, counted with multiplicity, in $\mathbb{P}^{1}_{\mathbb{L}}$. Thus the collections $\mathrm{Crit}(N)\cup\{\infty\}$ is a finite collection of elements in $\mathbb{P}^{1}_{\mathbb{L}}$, which sits in the Berkovich space $\mathbb{P}^{1}_{\mathrm{Ber}}$. Let $H_{\mathrm{big}}$ be convex hull of $\mathrm{Crit}(N)\cup\{\infty\}$. As a topological space, $H_{\mathrm{big}}$ is homeomorphic to the underlying space of a finite tree. We may equip this undelying space with a natural graph-theoretic combinatorial structure so that we may speak of vertices and edges. Consider the branch points as the vertices of $H_{\mathrm{big}}$. Then $H_{\mathrm{big}}$ is a finite tree and it plays a central role in our study. Since it is the biggest of several trees we will consider, we indicate this by the subscript.

In general, the vertices of $H_{\mathrm{big}}$ come in two flavors: (1) those in the hyperbolic space $\mathbb{H}_{\mathrm{Ber}}$, which are called the internal vertices of the tree, and (2) those in $\mathbb{P}^{1}_{\mathbb{L}}$, i.e. the critical points and the point $\infty$, which are called the leaves of the tree. Note the fixed points of $N_{r}$ in $\mathbb{P}^{1}_{\mathbb{L}}$ are $r_{i}$s and $\infty$. Then the convex hull $H_{\mathrm{fix}}$ of $\{r_{1},\cdots,r_{d},\infty\}$ sits in $H_{\mathrm{big}}$. As for $H_{\mathrm{big}}$, we can consider vertices, internal vertices and leaves for $H_{\mathrm{fix}}$. Then a vertex of $H_{\mathrm{fix}}$ is a vertex of $H_{\mathrm{big}}$. Hence $H_{\mathrm{fix}}$ is a subtree of $H_{\mathrm{big}}$. Let $V$ be the set of internal vertices of $H_{\mathrm{fix}}$. We have

[TABLE]

where $\mathrm{Val}_{E}(\xi)$ is the valance of $\xi\in E$ in a connected subset $E\subset\mathbb{P}_{\mathrm{Ber}}^{1}$, that is, the number of components of $E\setminus\{\xi\}$. Thus each element in $V$ is an internal vertex of $H_{\mathrm{big}}$. The convex hull $H_{V}$ spanned by $V$ is a subset of $\mathbb{H}_{\mathrm{Ber}}$. It also play a key role. The path distance metric on $\mathbb{H}_{\mathrm{Ber}}$ gives $H_{V}$ a natural metric structure that we study in section 5. Moreover, as we will discuss in [35], the combinatorial tree $H_{V}$ encodes the stratum in the Deligne-Mumford compactification of the moduli space $\mathcal{M}_{0,d+1}$ of Riemann surfaces of genus [math] with $d+1$ marked points to which the configuration $\{r_{1},\cdots,r_{d},\infty\}$ converges.

To make it clear, we list the subtrees we will study.

[TABLE]

where $V:=\{\xi\in H_{\mathrm{fix}}:\mathrm{Val}_{H_{\mathrm{fix}}}(\xi)\geq 3\}$.

To illustrate the these subtrees, we first define visible point, which was introduced by Faber [16]

Definition 3.8**.**

For a connected subset $E\subset\mathbb{P}^{1}_{\mathrm{Ber}}$ and a point $\xi\in\mathbb{P}^{1}_{\mathrm{Ber}}$, we say a point $\xi^{\prime}$ is the visible point from $\xi$ to $E$ if $\xi^{\prime}$ is the unique point of closure $\overline{E}$ of $E$ (in strong topology) minimize the small metric from $\xi$ to $E$. Denote by $\pi_{E}(\xi)$ the visible point from $\xi$ to $E$.

Note if $\xi\in\overline{E}$, we have $\pi_{E}(\xi)=\xi$. If $\xi\not\in\overline{E}$, then $\pi_{E}(\xi)$ is the unique point on $\overline{E}$ disconnecting $E$ from $\xi$.

Now we illustrate the several subtrees in $\mathbb{P}^{1}_{\mathrm{Ber}}$. As an motivating example, we consider

Example 3.9**.**

Let $r_{1}=0,r_{2}=t,r_{3}=2t,r_{4}=1,r_{5}=1+t$ and $r_{6}=2$ be six points in $\mathbb{L}$. Set $r=\{r_{1},r_{2},r_{3},r_{4},r_{5},r_{6}\}$ and consider the map $N_{r}$. Then $N_{r}$ has four free critical points $c_{1},c_{2},c_{3}$ and $c_{4}$. Figure 3 shows the convex hull $H_{\mathrm{big}}$ for $N_{r}$. The hull $H_{\mathrm{big}}$ has internal vertices $v_{1},v_{2},v_{3}\in\mathbb{H}_{\mathrm{Ber}}$ and leaves $r_{1},\cdots,r_{6},c_{1},\cdots,c_{4},\infty\in\mathbb{P}_{\mathbb{L}}^{1}$. Let $V=\{v_{1},v_{2},v_{3}\}$ and $H_{V}=\mathrm{Hull}(V)$. At $v_{1}$, the reduction of $N_{r}$ has degree $3$. Hence there are $4$ critical points of $N_{r}$ whose visible points to $H_{V}$ are $v_{1}$. At $v_{2}$, the map $(N_{r})_{\ast}$ has $1$ superattracting fixed point and $2$ attracting fixed points. Thus there are at least two directions whose corresponding Berkovich disks containing free critical points, say $c_{2},c_{4}$, attracted to these two attracting fixed points. At $v_{3}$, the reduction of $N_{r}$ has degree $2$.

For now, we focus on combinatorial aspects. The following theorem summarizes the properties of these subtrees. The proof depends on a sequence of lemmas established later in this subsection.

Theorem 3.10**.**

For a Netwon map $N(z)\in\mathbb{L}(z)$ of degree $d\geq 3$, consider the subtrees defined as above. Then we have

1. (1)

The set $V$ is a singleton if and only if the map $N$ has a potentially good reduction. 2. (2)

Let $\mathrm{Fix}_{\mathrm{Ber}}(N)$ be the set of fixed points of $N$ in $\mathbb{P}^{1}_{\mathrm{Ber}}$. Then

[TABLE] 3. (3)

The visible points from elements in $\mathrm{Crit}(N)$ to $H_{V}^{\infty}$ lie in $V$. Moreover, for each $v\in V$, there are $2\deg_{v}N-2$ points, counted with multiplicity, in $\mathrm{Crit}(N)$ whose visible points to $H_{V}^{\infty}$ are $v$. 4. (4)

For each $v\in V$, the visible points from the preimages $N^{-1}(v)$ to $H_{\mathrm{fix}}$ lie in $V$. 5. (5)

The set $H_{\mathrm{fix}}$ is forward invariant and contains $H_{V}^{\infty}$. Moreover, edges of $H_{\mathrm{fix}}$ maps to themselves. In particular, each point in $H_{r}\setminus H_{V}$ is attracted to some $r_{i}$. 6. (6)

*The Berkovich ramification locus $\mathcal{R}_{N}$ of $N$ consists of the union of closed segments *$[v,a]$s, where $a\in\mathrm{Crit}(N)$ and $v$ is the visible point from $a$ to $H_{V}$.

Remark 3.11**.**

In Theorem 3.10, the assumption $d\geq 3$ is necessary. Indeed, if $d=2$, then $H_{r}=H_{\mathrm{crit}}$ and $V$ is always a singleton. Hence $H_{V}^{\infty}$ is a segment and the ramification locus is connected which is $H_{r}$. But Theorem 3.10 $(2)-(5)$ are still true for $d=2$.

If $V=\{\xi\}$ is a singleton, then $\mathrm{Val}_{H_{\mathrm{fix}}}(\xi)=d$. Hence $\deg_{\xi}N=d$. Let $M\in\mathrm{PGL}_{2}(\mathbb{L})$ such that $M(\xi_{g})=\xi$. Then $\deg_{\xi_{g}}M^{-1}\circ N\circ M=d$. By Proposition 3.5, we have $\deg\mathrm{Red}(M^{-1}\circ N\circ M)=d$. Hence $N$ has a potentially good reduction. Conversely, if $N$ has a potentially good reduction, then there exits $M\in\mathrm{PGL}_{2}(\mathbb{L})$ such that $\deg\mathrm{Red}(M^{-1}\circ N\circ M)=d$. So $\deg_{\xi_{g}}M^{-1}\circ N\circ M=d$. Letting $\xi=M(\xi_{g})$, we have $\deg_{\xi}N=d$. Hence $V=\{\xi\}$ is a singleton. This gives the proof of Theorem 3.10 $(1)$.

Lemma 3.12**.**

$\pi_{H_{r}}(\infty)=\pi_{H_{V}}(\infty)$.

Proof.

Note

[TABLE]

Then $\mathrm{Val}_{H_{\mathrm{fix}}}(\pi_{H_{r}}(\infty))\geq 3$. Thus $\pi_{H_{r}}(\infty)\in V$. Hence $\pi_{H_{r}}(\infty)\in H_{V}$. Since $H_{V}\subset H_{r}$, we have

[TABLE]

The conclusion follows from the uniqueness of the visible point. ∎

For a rational map $\phi\in\mathbb{L}(z)$ and a type II point $\xi\in\mathbb{P}^{1}_{\mathrm{Ber}}(\mathbb{L})$, Proposition 3.2 claims that $\phi(\xi)=\xi$ if and only if the reduction $\mathrm{Red}(\psi)$ is nonconstant, where $\psi=M^{-1}\circ\phi\circ M$ and $M\in\mathrm{PGL}_{2}(\mathbb{L})$ such that $M(\xi_{g})=\xi$. A fixed point $\xi\in\mathrm{Fix_{Ber}}(\phi)$ is repelling if $\deg_{\xi}\phi\geq 2$. For the map $N$, the points $r_{1},\cdots,r_{d}$ and $\infty$ are the only fixed points of type I. Now we show

Lemma 3.13**.**

For the map $N$,

[TABLE]

In particular, the set of repelling fixed points of $N$ in $\mathbb{H}_{\mathrm{Ber}}$ is $V$.

Proof.

If $\xi\in H_{V}^{\infty}$ is a type II point, let $M\in\mathrm{PGL}_{2}(\mathbb{L})$ such that $M(\xi_{g})=\xi$. Then the reduction of $M^{-1}\circ N\circ M$ is nonconstant. Indeed, $M^{-1}\circ N\circ M\in\mathbb{L}(z)$ is a Newton map for a polynomial in $\mathbb{L}[z]$ with multiple roots. If $\xi\in H_{V}^{\infty}$ is not a type II point, there exists a sequence $\{\xi_{n}\}\subset H_{V\infty}$ of type II points such that $d(\xi_{n},\xi)\to 0$ as $n\to\infty$. Note $N$ is continuous. Then

[TABLE]

Thus $\xi\in\mathrm{Fix_{Ber}}(N)$.

If $\xi\not\in H_{V}^{\infty}$ is a type II point, let $M_{1}\in\mathrm{PGL}_{2}(\mathbb{L})$ such that $M_{1}(\xi_{g})=\xi$. Then there exist at least $d-1$ many $r_{i}$’s such that $\mathrm{Red}(M_{1}^{-1}(r_{i}))=\infty$. Thus, the reduction of $M_{1}^{-1}\circ N\circ M_{1}$ is constant. So $\xi\not\in\mathrm{Fix_{Ber}}(N)$. For any type III or IV point $\xi\not\in H_{V}^{\infty}$, the visible point $\eta:=\pi_{H_{\mathrm{fix}}}(\xi)$ is a type II point. Let $\vec{v}\in T_{\eta}\mathbb{P}^{1}_{\mathrm{Ber}}$ be such that $\xi\in\mathbf{B}_{\eta}(\vec{v})^{-}$. By Lemma 3.7, we know $N(\mathbf{B}_{\eta}(\vec{v})^{-})$ is a Berkovich disk. Since $\vec{v}$ is not a fixed point of the map $N_{\ast}$,

[TABLE]

So $\xi\not\in\mathrm{Fix_{Ber}}(N)$. Therefore, $\mathrm{Fix_{Ber}}(N)\cap\mathbb{H}_{\mathrm{Ber}}\subset H_{V}^{\infty}$.

Now we show that the set $V$ consisting of repelling fixed points in $\mathbb{H}_{\mathbb{L}}$. Let $\xi\in\mathrm{Fix_{Ber}}(N)$ be a type II point. Let $M_{2}\in\mathrm{PGL}_{2}(\mathbb{L})$ such that $M_{2}(\xi_{g})=\xi$. If $\xi\not\in V$, then $\mathrm{Red}(M_{2}^{-1}(r_{i}))$ is either $\infty$ or some constant $c\in\mathbb{C}$. Thus, the reduction $M_{2}^{-1}\circ N\circ M_{2}$ has degree $1$. Then by Proposition 3.5, $\deg_{\xi}N=1$. So $\xi$ is not a repelling fixed point. If $\xi\in V$, then there exist two distinct constants $c_{1}$ and $c_{2}$ in $\mathbb{C}$, and exist $r_{i}$ and $r_{j}$ with $i\not=j$ such that $\mathrm{Red}(M_{2}^{-1}(r_{i}))=c_{1}$ and $\mathrm{Red}(M_{2}^{-1}(r_{j}))=c_{2}$. Thus, the reduction $M_{2}^{-1}\circ N\circ M_{2}$ has degree at least $2$. Again, by Proposition 3.5, we have $\deg_{\xi}N\geq 2$. Thus $\xi$ is a repelling fixed point. ∎

Corollary 3.14**.**

$V=\{\xi\in H_{V}^{\infty}:\deg_{\xi}N\geq 3\}$.

Proof.

First note at $\infty$, the local degree $\deg_{\infty}N=1$. By the definition of $H_{V}$ and Lemma 3.13, we know

[TABLE]

For the other direction, let $\xi\in H_{V}^{\infty}\setminus V$. By Lemma 3.13, we have $\xi\in\mathrm{Fix_{Ber}}(N)$. Suppose $\xi$ is of type II. Let $M\in\mathrm{PGL}_{2}(\mathbb{L})$ be such that $M(\xi_{g})=\xi$. Then $M^{-1}\circ N\circ M$ has reduction of degree at most $1$. By Proposition 3.5, we have $\deg_{\xi}N\leq 1$. By the density of type II points and Lemma 3.3, we know $\{\xi\in H_{V}^{\infty}:\deg_{\xi}N\geq 3\}\subset V$. ∎

Recall that a point $c\in\mathbb{L}$ is a free critical point of $N$ if $c\in\mathrm{Crit}(N)\setminus\{r_{1},\cdots,r_{d}\}$. The following lemma gives the visible points of critical points to the tree $H_{V}^{\infty}$ and hence implies Theorem 3.10 $(3)$.

Lemma 3.15**.**

If $c\in\mathrm{Crit}(N)$ is a free critical point, then $\pi_{H_{V}^{\infty}}(c)\in V$. Moreover, for $v\in V$,

[TABLE]

counted with multiplicity.

Proof.

To ease the notations, let $\xi=\pi_{H_{V}^{\infty}}(c)$. Then $\xi\in\mathbb{P}^{1}_{\mathrm{Ber}}$ is a type II point. Let $\vec{v}\in T_{\xi}\mathbb{P}^{1}_{\mathrm{Ber}}$ be the tangent vector such that $c\in\mathbf{B}_{\xi}(\vec{v})^{-}$. Then the directional multiplicity $m_{N}(\xi,\vec{v})\geq 2$ since $c$ is a critical point of $N$. Thus by Lemma 3.4, we have

[TABLE]

By Corollary 3.14, we have $\xi\in V$.

For $v\in V$, by the definition of $V$ and Lemma 3.13, we know $v$ is a type II fixed point of $N$. Let $M\in\mathrm{PGL}_{2}(\mathbb{L})$ such that $M(\xi_{g})=v$. Then the reduction $\mathrm{Red}(M^{-1}\circ N\circ M)$ has $2\deg_{v}N-2$ critical points. Note

[TABLE]

and

[TABLE]

Thus we obtain the conclusion. ∎

Now we prove a weak version of Theorem 3.10$(4)$ which allows us to characteristic the fixed Berkovich Fatou components of $N$ and hence prove Theorem 3.10$(5)$. After that, we prove Theorem 3.10$(4)$.

Lemma 3.16**.**

For $v\in V$, let $\xi\in N^{-1}(v)$. Then

[TABLE]

Moreover, for $v_{1}\not=v_{2}\in V$,

[TABLE]

Proof.

Suppose $\eta:=\pi_{H_{V}}(\xi)\in H_{V}\setminus V$. By Lemma 3.13 and Corollary 3.14, we have $\eta\in\mathrm{Fix_{Ber}}(N)$ and $\deg_{\eta}N=1$. Let $\vec{v}\in T_{\eta}\mathbb{P}^{1}_{\mathrm{Ber}}$